1. Prove that the sets {∨, ¬} and {→, ¬} are
sufficient, i.e., that all other propositional connectives can
be described in terms of these connectives.

2. Use natural deduction to prove that p → q |−
¬p ∨ q.

3. Use resolution to prove that ¬p |− p → q.

4. Use program synthesis to solve the Archimedes problem re
whether the crown was made of gold or of silver. We know that
m1 = ρ1 * V1, that
m2 = ρ2 * V2, and that
based on the values ρ1 and ρ2, we
can tell whether the crown was made of gold or of silver:

if ρ1 = ρ2, this means that
the crown was made of gold, and

if ρ1 ≠
ρ2, this means that the crown was made of
silver.

Assume that we know m1, m2,
V1, and V2. Synthesize a program that,
based on this data, checks whether the crown was made of gold
or of silver.

5. (For extra credit) Prove that there exists an irrational
number x for which the power x√5 is
rational.